Like many here, I don't find the argument that convincing.
The original argument is:
(1) If there are no objective values then there are no facts (since there’s nothing that we ought to believe).
(2) There are facts.
(C) Therefore there are objective values.
(2) seems just fine, and most folk who deny the existence of objective values will accept (2), so probably not much need for an argument there. The trouble all starts with (1). What exactly is the argument for (1)?
I think the argument for (1) is meant to be the part in the brackets. Looking through your discussion with Terrapin, I guess you had this in mind:
(3) Facts are things we ought to believe.
(4) If there are no objective values then there is nothing we ought to believe.
(C2) If there are no objective values then there are no facts.
And (C2) here is identical with (1), giving us an argument for (1). But there are problems with this argument. The inference from (3) and (4) to (C2) is only valid if (3) is meant as a definition of "facts". If "fact" is defined as "something we ought to believe", it then follows that if there are no objective values there are no facts. But if (3) is instead taken to mean "we ought to believe facts", where "fact" means something like "propositions stating how things are", the argument will be fallacious. In that case, that there are no objective values and nothing we ought to believe will not entail that there are no propositions stating how things are (facts), since (and this was pointed out by others) it might be that there is a way things are without there being any rules about how we should/should-not believe.
So we must take (3) as a definition of "fact". The problem is, if we take (3) as a definition, no one who denies or is agnostic about the existence of objective values will accept your
(2) without argument. That is, they will not accept that there are facts, because they will not accept that there are things we ought to believe. And all this is just to say that the argument will not convince anyone who doesn't already agree that there are objective values, making it a poor argument indeed.
Now, you did offer a different argument, which was to claim that denying that there are things we ought to believe is "absurd". You look to prove that it is absurd by this argument:
Any claim that there are no facts (nothing that we ought to believe) can be met with the questions, “Is that a fact? Ought we to believe that?” and so on to infinity. — AJJ
and again in talking with Wayfarer:
It is absurd to say facts are not things we ought to believe. Because every denial invites the question, “Well ought we to believe that? Ought we to believe that facts are not things we ought to believe? If the answer is no, well ought we to believe that? And so on. It’s really just this I’m appealing to to demonstrate that facts definitely are things we ought to believe, because we ought to believe true things, which are facts. — AJJ
So any person who denies that there are things we ought to believe can be asked
"well, ought we believe that there is nothing we ought to believe?", and the sceptic will have to reply
"no, there is nothing we ought to believe", which puts him, according to you, in an absurd position. I don't see why it is absurd. He denies that there is anything which we ought to believe, and, consistently, holds that even his own assertions are not such that we ought to believe them. We may choose to believe his claims or not, as we please. This seems a perfectly coherent position. I can describe my own beliefs without insisting that others ought to have those beliefs, and so can the sceptic. He can say that there is nothing we ought to believe, without insisting that this is something we ought to believe. He might nevertheless be correct. I don't see why the mere fact that you could ask him the question again does anything to hurt his position. It doesn't show that his position is false or contradictory, nor is there any barrier to understanding his position. The regress doesn't seem to be vicious.
PA